In this paper, with the help of one-loop perturbative RG analysis in \(d=4-\epsilon \) ,
we have studied the fermionic quantum criticality with enlarged Ising\(\times \) Ising fluctuation
in the two dimensional honeycomb materials.
To get an understanding of whether the semimetal-insulator transition is weak first-order or second-order
transition, we include a cubic term of the order-parameter in the theory and study its fate as the infrared stable fixed-point is approached.
The semimetal-CDW transition is modeled in terms of an Ising GNY theory with a generalized flavors of Dirac fermions \(N_f\) ,
we find the cubic term is always irrelevant if \(N_f\)  fulfills \(N_f\ge 1/4\)  in three space-time dimensions. The irrelevance implies that the extra fluctuations from fermions change of the nature of transition and render it continuous[1]}, [2]}, [3]}.
We also calculate the complete second-order regime for varying space-time and \(N_f\) , as shown in Fig. REF .
